# Knot invariant with multiple skein relations

## TL;DR

This paper introduces a new link invariant constructed via multiple skein relations and crossing smoothings, generalizing the HOMFLYPT and Kauffman polynomials, with simplified computation methods.

## Contribution

It presents novel smoothing techniques and a skein system to create a more powerful, computable link invariant that extends existing polynomial invariants.

## Key findings

- Invariant generalizes HOMFLYPT polynomial
- Invariant extends two-variable Kauffman polynomial
- Simplified version is easy to compute

## Abstract

Given any oriented link diagram, one can construct knot invariants using skein relations. Usually such a skein relation contains three or four terms. In this paper, the author introduces several new ways to smooth a crossings, and uses a system of skein equations to construct link invariant. This invariant can also be modified by writhe to get a more powerful invariant. The modified invariant is a generalization of both the HOMFLYPT polynomial and the two-variable Kauffman polynomial. Using the diamond lemma, a simplified version of the modified invariant is given. It is easy to compute and is a generalization of the two-variable Kauffman polynomial.